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@@ -45,20 +45,15 @@ From a more mathematical perspective, the techniques developed for approximating
 Besides theoretical interests, \scut{} is useful in practical scenarios such as in image segmentation and in some machine leaning algorithms.
 
 \subsection{related works}
-\nonuscut{} is APX-hard \cite{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\cite{chawla_hardness_2005}. \scut{} admits no PTAS \cite{uniformhardnessFocs07}, assuming a widely believed conjecture. The currently best approximation algorithm for \scut{} has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \cite{arora_osqrtlogn_2010}. Prior to this currently optimal result, there is a long line of research optimizing both the approximation ratio and the complexity, see \cite{arora_expander_2004,leighton_multicommodity_1999}.
+\nonuscut{} is APX-hard \cite{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\cite{chawla_hardness_2005}. \scut{} admits no PTAS \cite{uniformhardnessFocs07}, assuming that NP-complete problems cannot be solved in randomized subexponential time. The currently best approximation algorithm for \scut{} has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \cite{arora_osqrtlogn_2010}. Prior to this currently optimal result, there is a long line of research optimizing both the approximation ratio and the complexity, see \cite{arora_expander_2004,leighton_multicommodity_1999}.
 There are also works concerning approximating \scut{} on special graph classes such as planar graphs \cite{lee_genus_2010}, graphs with low treewidth \cite{chlamtac_approximating_2010,gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}.
 
-For an overview of the LP methods for \scut{}, see \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}.
+The seminal work of \cite{leighton_multicommodity_1999} starts this line of research. They studied multicommodity flow problem and proved a $O(\log n)$ flow-cut gap (in fact the tight $\Theta(\log n)$ gap was proven by \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995}). They also developed $O(\log n)$ approximation algorithm for multicommodity flow problems, which implies $O(\log n)$ approximation for \scut{}. The technique is called region growing. They also discovered a lowerbound of $\Omega(\log n)$ via expanders. Note that any algorithm achieving the $O(\log n)$ flow cut gap implies an $O(\log^2 n)$ approximation for \nonuscut{}, but better ratio is still possible through other methods. This paper showed that $O(\log^2 n)$ is the best approximation we can achieve using flow-cut gap.
 
-% \subsection{open problems}
+For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees an approximation ratio of $O(\log^2 n)$. This is further improved by \cite{Linial_London_Rabinovich_1995} and \cite{lognGapAumann98}. \cite{lognGapAumann98} applied metric embedding to \nonuscut{} and obtained a $O(\log n)$ approximation. The connections between metric embedding and \nonuscut{} is influential. \nonuscut{} can be formulated as an integer program. \cite{lognGapAumann98}, \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995} considered the metric relaxation of the IP. They observed that \nonuscut{} is polynomial time solvable for trees and more generally for all $\ell_1$ metrics. The $O(\log n)$ approximation follows from the $O(\log n)$ distortion in the metric embedding theorem.
-
-% One major open problem for \scut{} is the best approximation ratio for planar graphs. It is conjectured that the ratio for planar graphs is $O(1)$ but currently the best lowerbound is $O(\sqrt{\log n})$. For graphs treewidth $k$, an open problem is that whether there is a 2 approximation algorithm that runs in $2^{O(k)}\poly(n)$.
-
-The seminal work of \cite{leighton_multicommodity_1999} starts this line of research. They studied multicommodity flow problem and proved a $O(\log n)$ flow-cut gap (in fact the tight $\Theta(\log n)$ gap was proven by \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995}). They also developed $O(\log n)$ approximation algorithm for multicommodity flow problems, which can imply $O(\log n)$ approximation for \scut{} and $O(\log^2 n)$ approximation for \nonuscut{}. The technique is called region growing. They also discovered a lowerbound of $\Omega(\log n)$ via expanders. Note that any algorithm achieving the $O(\log n)$ flow cut gap implies an $O(\log^2 n)$ approximation for \nonuscut{}, but better ratio is still possible through other methods. This paper showed that $O(\log^2 n)$ is the best approximation we can achieve using flow-cut gap.
-
-For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees a $O(\log^2 n)$ approximation. This is further improved by \cite{Linial_London_Rabinovich_1995} and \cite{lognGapAumann98}. \cite{lognGapAumann98} applied metric embedding to \nonuscut{} and obtained a $O(\log n)$ approximation. The connections between metric embedding and \nonuscut{} is influential. \nonuscut{} can be formulated as an integer program. \cite{lognGapAumann98}, \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995} considered the metric relaxation of the IP. They observed that \nonuscut{} is polynomial time solvable for trees and more generally for all $\ell_1$ metrics. The $O(\log n)$ approximation follows from the $O(\log n)$ distortion in the metric embedding theorem.
 
 \cite{arora_expander_2004} and \cite{arora_osqrtlogn_2010} further improved the approximation ratio for \scut{} to $O(\sqrt{\log n})$ via semidefinite relaxation. This is currently the best approximation ratio for \scut{}.
+For \nonuscut{}, the approximation is improved to $O(\sqrt{\log n} \log \log n)$ \cite{arora_euclidean_2005,arora_frechet_2007}. Later \cite{guruswami_approximating_2013} gives a $\frac{1+\delta}{\e}$ approximation in time $2^{r/(\delta \e)}\poly(n)$ provided that $\lambda_r\geq \opt / (1-\delta)$.
 
 There is also plenty of research concerning \scut{} on some graph classes, for example \cite{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \cite{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \cite{Cohen-Addad_Mömke_Verdugo_2024} obtained
 a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running time $2^{2^{O(k)}}\poly(n)$.

ref.bib

@@ -243,3 +243,50 @@ numpages = {10},
 location = {San Francisco, California, USA},
 series = {SODA '95}
 }
+
+@article{arora_frechet_2007,
+	title = {Fréchet {Embeddings} of {Negative} {Type} {Metrics}},
+	volume = {38},
+	issn = {1432-0444},
+	url = {https://doi.org/10.1007/s00454-007-9007-0},
+	doi = {10.1007/s00454-007-9007-0},
+	language = {en},
+	number = {4},
+	urldate = {2025-05-19},
+	journal = {Discrete \& Computational Geometry},
+	author = {Arora, Sanjeev and Lee, James R. and Naor, Assaf},
+	month = dec,
+	year = {2007},
+	keywords = {Convex and Discrete Geometry, Differential Geometry, Distortion, Euclidean, Functional Analysis, Geometry, Global Analysis and Analysis on Manifolds, Hyperbolic Geometry, L 1, Metric embeddings, Sparsest cut problem},
+	pages = {726--739},
+}
+
+@inproceedings{arora_euclidean_2005,
+	address = {New York, NY, USA},
+	series = {{STOC} '05},
+	title = {Euclidean distortion and the sparsest cut},
+	isbn = {978-1-58113-960-0},
+	url = {https://doi.org/10.1145/1060590.1060673},
+	doi = {10.1145/1060590.1060673},
+	urldate = {2025-05-19},
+	booktitle = {Proceedings of the thirty-seventh annual {ACM} symposium on {Theory} of computing},
+	publisher = {Association for Computing Machinery},
+	author = {Arora, Sanjeev and Lee, James R. and Naor, Assaf},
+	month = may,
+	year = {2005},
+	pages = {553--562},
+}
+
+@inproceedings{guruswami_approximating_2013,
+	address = {USA},
+	series = {{SODA} '13},
+	title = {Approximating non-uniform sparsest cut via generalized spectra},
+	isbn = {978-1-61197-251-1},
+	urldate = {2025-05-18},
+	booktitle = {Proceedings of the twenty-fourth annual {ACM}-{SIAM} symposium on {Discrete} algorithms},
+	publisher = {Society for Industrial and Applied Mathematics},
+	author = {Guruswami, Venkatesan and Sinop, Ali Kemal},
+	month = jan,
+	year = {2013},
+	pages = {295--305},
+}